I would like everything to be symbolic. If you have two reference frames, say A and B, you can only take the dot product of a vector expressed using the A coordinates with a vector expressed in the B coordinates if you know the rotation matrix that relates the two reference frames. For example: Take A and B to be two reference frames that have a set of 3 dextral orthonormal vectors ( i.e, dot(a1,a2)=dot(a2,a3)=dot(a1,a3)=0, cross (a1,a2)=a3, cross(a2,a3)=a1, and cross(a3,a1)=a2). Define the orientation of B relative to A by initially aligning a1 with b1, a2 with b2, a3 with b3, then performing a right handed rotation of B relative to A by an angle q1 about an axis parallel to a3 (and b3). Then: b1 = cos(q1)*a1 + sin(q1)*a2 b2 = -sin(q1)*a1 + cos(q1)*a2 b3 = a3
Normally I would write a1,a2,a3,b1,b2,b3 in bold or with a carat above them, the way you might see them in a physics class or a dynamics class, but I think you get the idea. Now, in order to take a dot product of two vectors that are expressed in the same reference frame coordinates, we use the following rules: dot(a1,a2) = 0 dot(a1,a3) = 0 dot(a2, a3) = 0 dot(a1,a1)=1 dot(a2,a2)=1 dot(a3,a3)=1 So any vector expression that only involves terms with a1,a2,a3, we can easily a dot() function to implement the above rules for the dot product. But what if we need to dot multiply a1 and b1? Then we need to either 1) express a1 in terms of B coordinate basis vectors, or 2) express b1 in terms of A coordinate basis vectors. Here is where we need to rotation matrix defined above, and here is how I'm not sure of how to best implement things. I'm looking for a good way to store the rotation matrices with the reference frame objects.... If one defines successive rotations, say, starting with A, then orienting B relative to A, then C relative to B, then D relative to C, we want to have the dot() function be able to resolve the dot product of say d1 and a1, even though the rotation matrix between D and A wasn't explicitly defined (instead is it is implicitly defined through a series of matrix multiplications). One could imagine a long chain of rotations, with various branches (as is common in the case of multi-body systems). How to best store the and resolve the various rotations is what I'm after. The reason I would like to do this is to be ably to use the following syntax to define the position of one point q relative to another p: p_q_p = q2*a1 + q3*b2 and the be able to do: dot(p_q_p,a1) and get: q2 - sin(q1)*q3 (recall, b2 = -sin(q1)*a1+cos(q1)*a2) Hopefully this clarifies what I'm trying to do. Thanks, ~Luke On Jan 17, 5:12 am, Alan Bromborsky <abro...@verizon.net> wrote: > Luke wrote: > > I'm trying to figure out how I could implement in Sympy a way to take > > dot and cross products of unit vectors in different reference frames. > > For example, suppose I have a reference frame 'A', along with 3 > > dextral orthonormal vectors: a1, a2, a3. I think I can figure out how > > to code up something so that dot and cross products in the same > > reference frame would give the following results: > > In[1]: dot(a1,a2) > > Out[1]: 0 > > In[2]: dot(a1,a1) > > Out[2]: 1 > > In[3]: cross(a1,a2) > > Out[3]: a3 > > > I haven't done this yet, but I think it should be pretty > > straightforward. What I would like to extend this to is to be able to > > dot and cross multiply unit vectors from different reference frames. > > For example, suppose I introduce a second reference frame 'B', along > > with 3 dextral orthonormal vectors: b1, b2, b3, and I orient B by > > aligning B with A then performing a simple rotation of B about the > > a3=b3 axis by an angle q1. Then: > > b1 = cos(q1)*a1 + sin(q1)*a2 > > b2 = -sin(q1)*a1 + cos(q1)*a2 > > b3 = a3 > > > I would like to then be able to do: > > In[4]: dot(a1,b1) > > Out[4]: cos(q1) > > > etc... > > > I guess what I'm not sure about is how to structure all the objects so > > that if a rotation matrix hasn't been defined between two reference > > frames, an exception is raised and you can't take the dot or cross > > product. It would be ideal to be able to handle long chains of simple > > rotations so that every possible rotation matrix wouldn't need to be > > defined explicitly, i.e., if I specify the orientation of B relative > > to A (as above), and then the orientation of another reference frame C > > relative to B, and then try to take a dot product between a1 and c1, > > the two rotation matrices (C to B, B to A) get multplied and then used > > to compute the dot product. > > > I'm familiar with classes and python fairly well, and I know the > > kinematics well, but I'm by no means an experience object oriented > > programmer, so I'm not sure about how the best way to structure things > > would be. > > > Any suggestions on how I might start on something like this? > > > Thanks, > > ~Luke > > Do you want to do this purely symbolically or are you going to put in > specific values? That is do you want sympy only to generate symbolic > formulas that you will later use in a program or do you want sympy to > also do the numerical evaluation? --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "sympy" group. To post to this group, send email to sympy@googlegroups.com To unsubscribe from this group, send email to sympy+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sympy?hl=en -~----------~----~----~----~------~----~------~--~---
